How to shoot a character from one platform to another. (BGE)
Final result:



What's happening

So I have a player, a floor for the player to walk on, a wall to jump over, and a cannon to launch him.

The player has a basic setup:


W to move forward, A to turn left, S to move backward, and so on.
The motion of the Dynamic player is simply the Location of the Motion Actuator.
(You probably have a different setup)

Now, inside the cannon, I added a child cube, that I'm calling inside. It's static and has the cannon as a parent. It's also Invisible and has the Ghost property applied.



The logic bricks

Referencing the above image, when the player comes into contact with the inside cube, the keyboard sensors will activate. When A is pressed, they will activate the cannon's Motion Actuator, that will turn the cannon 2 degrees along the Global Z axis. Likewise, when D is pressed, it will move -2 degrees. 

Last of all, when the Spacebar is pressed while in contact with the inside cube, the Player's Motion Actuator is triggered with a force of 800 along the Z axis. Because the cannon is slanted, the player will hit the top of the cannon and will be forced out of the cannon at the same angle. This will not work if your cannon is vertical.

Here is the .blend file: